The Rayleigh–Bénard problem for water with maximum density effects

Abstract

Linear stability and weakly nonlinear stability analyses are developed for Rayleigh–Bénard convection in water near 3.98 °C subject to isothermal boundary conditions. The density–temperature relationship (equation of state) is approximated by a cubic polynomial, including linear, quadratic, and cubic terms. The continuity equation, the Navier–Stokes momentum equation, the equation of state, and the energy equation constitute the governing system. Linear stability analysis is used to investigate how the maximum density property of water affects the onset of convective instability and the choice of unstable wave number for four different types of boundary conditions. Then, a weakly nonlinear stability study is done using the spectral Fourier method for isothermal tangential stress-free boundary conditions to quantify the heat transport of the system and demonstrate the transition from regular/periodic convection to chaotic convection. A Stuart-Ginzburg–Landau equation is obtained using the multiscale expansion method. Streamlines and isotherms are presented and analyzed. The influence of maximum density has been shown to delay the onset of instability and is, therefore, a stabilizing mechanism for thermal instability. Due to the maximum density, the onset of chaotic convection is also delayed. Among four different boundaries, the impermeable rigid boundaries require the highest Rayleigh number for instability to begin. Increasing boundary temperatures advance the onset of chaotic convection and improve the heat transport situation.